Question: Factor the following expression: $4$ $x^2$ $-5$ $x+$ $1$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(4)}{(1)} &=& 4 \\ {a} + {b} &=& & & {-5} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $4$ and add them together. The factors that add up to ${-5}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${-4}$ $ \begin{eqnarray} {ab} &=& ({-1})({-4}) &=& 4 \\ {a} + {b} &=& {-1} + {-4} &=& -5 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {4}x^2 {-1}x {-4}x +{1} $ Group the terms so that there is a common factor in each group: $ ({4}x^2 {-1}x) + ({-4}x +{1}) $ Factor out the common factors: $ x(4x - 1) - 1(4x - 1) $ Notice how $(4x - 1)$ has become a common factor. Factor this out to find the answer. $(4x - 1)(x - 1)$